"PHILOSOPHICALLY SPEAKING...":

This article introduces paraconsistent logic, the idea, borrowed from formal mathematics, that paradoxes can be accepted into an argument rather than necessarily eliminated or resolved for an argument to proceed.

"Suppose you are waiting for a friend. They said they would meet you around 5pm. Now it is 5:07. Your friend is late. But then again, it is still only a few minutes after 5pm, so really, your friend is not late yet. Should you call them? It is a little too soon, but maybe it isn’t too soon … because your friend is both late and not late. (What they’re not is neither late nor not late, because you are clearly standing there and they clearly haven’t arrived.) ... Often, there are two or more competing theories to explain some given data. How do we decide which to adopt? A standard account from Thomas Kuhn, in 1977, is that we weigh up various theoretical virtues: consistency, yes, but also explanatory depth, accord with evidence, elegance, simplicity, and so forth. Ideally, we might have all of these, but criteria such as simplicity will be set aside if it is outweighed by, say, predictive power. And so too for consistency, say paraconsistent logicians such as Priest and Sylvan. Any of the theoretical virtues are virtuous only to the extent that they match the world. For example, all else being equal, a simpler theory is better than a more complicated one. But ‘all else’ is rarely equal, and as people from Aristotle to David Hume point out, the simpler theory is only better to the extent that the world itself is simple. If not, then not. So too with consistency. The virtue of any given theory then will be a matter of its match with the world. But if the world itself is inconsistent, then consistency is no virtue at all. If the world is inconsistent – if there is a contradiction at the bottom of logic, or at the bottom of a bowl of cereal – a consistent theory is guaranteed to leave something out."
aeon.co/essays/paraconsistent-logics-find-structure-in-our-inconsistent-world